A group of gears is defined as a set of 2-dinemsional disks with the teeth being ignored. Each gear can only rotate either clockwise or anti-clockwise according to a fixed axes. The gears are not overlapping each other. If one gear is rotating clockwise, then another one which is tangent to it will rotate anti-clockwise, and vice versa. Here we assume that the gears will never skid.
Given the centers and radiuses of n gears which are indexed from 1 to n. One questions is: will the n-th gear be driven by the 1st gear if we rotate the 1st one clockwise?
In Figure 1, the 1st gear is stuck so it cannot be rotated at all. In Figure 2, the n-th (3rd) gear is not tangent to any of the other gears and hence will not be driven by the 1st one.
If it is possible to drive gear n by gear 1, then let us rotate gear 1 clockwise with a steady speed. Assume that there is a worm initially sticking at the left-most point (the 2-D point with the smallest x coordinate) on gear 1. When the worm is driven to a tangent point to another gear, it can jump to the other gear or stay where it is. What is the minimum distance that worm must travel, if it is to reach the right-most point on gear n? Please keep in mind that the worm can be driven by a gear only, not by itself.
Input
The input consists of several test cases. In each test case:
the first line contains an integer n (2 <= n <= 20), which is the number of gears;
and the following n lines gives the information of the n gears, each occupies one line. To be more specific, in the i-th line, there are xi and yi, the coordinates of the i-th disk center, and the radius ri of the i-th disk.
Note:
xi, yi, and ri are all floating point numbers.
Two disks (x1, y1, r1) and (x2, y2, r2) are considered to be tangent to each other, if fabs(r1 + r2 - dist) < 1e-5, where dist is the distance between their centers.
The input is finished by n = -1.
Output
For each test case, if gear n cannot be driven by gear 1, print "Wrong combination"; otherwise print the minimum distance that the worm must move, up to 3 decimal places.
Sample Input
3
0 0 1
2 0 1
1 1.7320508 1
4
0 0 1
2 0 1
0 2 1
2 2 1
-1
Sample Output
Wrong combination
6.283
